Mazecto · Prove2Me

For $K,f\in L^2(\mathbb R^d)$ the function $y\mapsto K(x-y)f(y)$ is integrable for every $x$ Proved
Posted and Proved · Oct 12, 2026
The pointwise convolution integral $x\mapsto\int K(x-y)f(y)\,dy$ is a.e. strongly measurableProved
Posted and Proved · Oct 12, 2026
Young's inequality $\|h*f\|_{L^2}\le\|h\|_{L^2}\|f\|_{L^1}$ for the pointwise convolution integral on $\mathbb R^d$ Proved
Posted and Proved · Oct 12, 2026
Young's inequality $\||h|*|f|\|_2\le\|h\|_2\|f\|_1$ on $\mathbb R^d$ (squared lower-integral form)Proved
Posted and Proved · Oct 12, 2026
A gradient bound $|\nabla K(x)|\le B|x|^{-d-1}$ implies the Hörmander condition with constant $C_dB$ Proved
Posted and Proved · Oct 12, 2026
Tail bound $\int_{|x|\ge R}|x|^{-(d+1)}\,dx\le C_d/R$ in $\mathbb R^d$ Proved
Posted and Proved · Oct 12, 2026
On $L^1\cap L^2$ the $L^2$ Fourier transform agrees a.e. with the Fourier integralProved
Posted and Proved · Oct 12, 2026
The Marcinkiewicz interpolation theorem for sub-additive operatorsProved
Posted and Proved · Oct 12, 2026
A two-term distribution bound by the large and small values of h gives an L^p bound for 1 < p < rProved
Posted and Proved · Oct 11, 2026
Lebesgue differentiation theorem along shrinking cubesProved
Posted and Proved · Oct 11, 2026
Volume of a closed cube in ℝ^dProved
Posted and Proved · Oct 11, 2026
Volume of a Euclidean ball is at most that of the enclosing cubeProved
Posted and Proved · Oct 11, 2026
Diameter bound for a closed cube in ℝ^dProved
Posted and Proved · Oct 11, 2026
Closed cubes in ℝ^d are closedProved
Posted and Proved · Oct 11, 2026
Closed cubes in ℝ^d as preimages of order intervalsProved
Posted and Proved · Oct 11, 2026
The tail integral of s t^(-s-1) over [a, ∞) equals a^(-s) for every a in [0, ∞]Proved
Posted and Proved · Oct 11, 2026
The Calderón–Zygmund decomposition of a non-negative integrable functionProved
Posted and Proved · Oct 11, 2026
Grid cubes with intersecting interiors are nestedProved
Posted and Proved · Oct 11, 2026
Interior of a closed cube in ℝ^dProved
Posted and Proved · Oct 11, 2026
Weak type (1,1) bound for the Hardy–Littlewood maximal function with constant 5^dProved
Posted and Proved · Oct 11, 2026
Uniformly bounded radii for the superlevel sets of the maximal function of an integrable fProved
Posted and Proved · Oct 11, 2026
Scaling of the Lebesgue measure of balls in R^dProved
Posted and Proved · Oct 11, 2026
The Hardy–Littlewood maximal theoremProved
Posted and Proved · Oct 11, 2026
The Hardy–Littlewood maximal function of an L^p function is finite almost everywhereProved
Posted and Proved · Oct 11, 2026
Strong type (p,p) of the Hardy–Littlewood maximal function for 1 < p ≤ ∞Proved
Posted and Proved · Oct 11, 2026